Pythagoras' Theorem and Trigonometry

OCR

GCSE

This topic unifies geometric reasoning with algebraic manipulation through the application of Pythagoras' Theorem and Trigonometry. Candidates must demonstrate AO1 fluency in selecting and applying $a^2 + b^2 = c^2$ and trigonometric ratios (SOH CAH TOA) to determine missing dimensions in right-angled triangles. Higher-order assessment (AO3) requires the extension of these principles into 3D coordinate systems, the derivation of exact values, and the analysis of non-right-angled triangles using the Sine and Cosine rules. Mastery is evidenced by the ability to construct geometric models from complex worded contexts and resolve forces or bearings with precision.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

What You Need to Demonstrate

Key skills and knowledge for this topic

Award M1 for explicitly stating the correct trigonometric ratio or Pythagorean formula selected before substitution
Award M1 for correct substitution of known values into the formula (e.g., sin(30) = x/12)
Award A1 for the final answer correct to the specified degree of accuracy (usually 3 significant figures)
In 'Show that' questions, award marks only when the full chain of reasoning is visible; implicit calculation steps are not credited
Award B1 (Higher Tier) for correct recall of exact trigonometric values (e.g., tan 45 = 1) without calculator evidence

Marking Points

Key points examiners look for in your answers

Award M1 for explicitly stating the correct trigonometric ratio or Pythagorean formula selected before substitution
Award M1 for correct substitution of known values into the formula (e.g., sin(30) = x/12)
Award A1 for the final answer correct to the specified degree of accuracy (usually 3 significant figures)
In 'Show that' questions, award marks only when the full chain of reasoning is visible; implicit calculation steps are not credited
Award B1 (Higher Tier) for correct recall of exact trigonometric values (e.g., tan 45 = 1) without calculator evidence

Examiner Tips

Expert advice for maximising your marks

💡Always write 'SOH CAH TOA' at the top of the page to prevent ratio selection errors under pressure
💡For 3D problems (Higher Tier), explicitly sketch the 2D right-angled triangle slice you are working on to isolate the relevant variables
💡When the question asks for an 'exact value', do not use a calculator approximation; leave the answer in surd form or as a fraction
💡Check the question for specific rounding instructions (e.g., 'to 1 decimal place'); default to 3 significant figures if not specified

Common Mistakes

Pitfalls to avoid in your exam answers

Squaring the hypotenuse and adding a shorter side when finding a missing shorter side in Pythagoras problems
Calculator set to Gradients or Radians mode, resulting in incorrect numerical values for trigonometric ratios
Premature rounding of intermediate values (e.g., rounding 1.414 to 1.4) leading to a final answer outside the accepted accuracy range
Confusing the selection of Sine, Cosine, and Tangent ratios due to incorrect labelling of Adjacent and Opposite sides relative to the angle

Key Terminology

Essential terms to know

Pythagoras' Theorem in 2D and 3D contexts

Trigonometric ratios (SOH CAH TOA) in right-angled triangles

The Sine Rule, Cosine Rule, and Area of a Triangle ($1/2 ab \sin C$)

Exact trigonometric values ($0, 30, 45, 60, 90$) and surd manipulation

Angles of elevation, depression, and three-figure bearings

Likely Command Words

How questions on this topic are typically asked

Calculate

Show that

Work out

Find

Give

Practical Links

Related required practicals

•{"code":"Navigation","title":"Bearings and Distances","relevance":"Calculating direct distances between points given bearings"}
•{"code":"Construction","title":"Pitch and Inclination","relevance":"Determining roof angles or ramp gradients"}

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